Lectura de la tesis de  Manuel Mellado Cuerno

Sala 520, viernes 9 de febrero 11:30.

Título de la tesis: On isometric embeddings of metric spaces.

Director de la tesis: Luis Guijarro Santamaría.

Abstract: Extracting and describing geometric and topological properties of certain metric spaces is often a challenging and costly task. Therefore, employing techniques that do not directly engage with these objects is often a wise strategy. One of the most commonly used approaches involves embedding these spaces into other spaces known as ambient spaces. The goal is to work with isometric embeddings: the distance function of the ambient space restricted to the image of the embedding of our initial space must coincide with the distance function of the original metric space.

In this talk, we will study two specific isometric embeddings: the Kuratowski embedding and the canonical embedding into Wasserstein-type spaces.

Regarding the first one, we will show some upper and lower bounds for the Filling Radius (an invariant presented by Gromov) showing the universal positivity of the invariant an bounds for Riemannian submersions and submetries. After this, we will connect the Kuratowski embedding and the one into Wasserstein-type spaces via the reach (in the sense of Federer). We will display results concerning the reach of the image of isometric embeddings into those type of spaces. Finally, we will study The Rival Coffee Shop Problem, an applied problem where the Wasserstein distance gives a new approach to solve it.